Factual error: When the archangels jump from the helicopter, the "bait" is looking behind him saying he can see the dragon. For a parajumper to look between his legs to view behind himself, he would de-arch and thus de-stabilise himself while travelling at that speed. It would send the jumper into a barrel-roll spin.
Plot hole: A starving, hundred tonne creature incinerates Quinn's little veggie patch, then stops to devour the ash. Later Quinn comes back and finds the tomato vines blackened but still there, their stakes intact (not even knocked down), and even some crisp but still edible fruit on the vines.
Continuity mistake: I've seen it in other films but in this one it gets almost ridiculous. Van Zan's beard changes from pretty thick when he's introduced to almost shaved the same day after the first dragon fight then back to thick and fat a couple of days later while in London. Even if we believe he shaved after the dragon fight - which shouldn't be the case since the camera follows him almost the whole time - there's no way it got that fat 2 days later.
Plot hole: Given that the dragons overwhelmed the worlds' militaries, the presented ways to bring them down are very gentle, so to speak. The force of impact on the ground would easily be achieved by an air to air or surface to air missile, or an RPG. The crossbow with some explosives has far less strength than a tank shell (weak points or no, the difference is just too great). The dragons are quite tough individually, but would have to succumb to strikes from modern weapons, or even to improvised ancient weapons of modern make. Humans that are able to keep helicopters and M1A2 tanks in running condition would be able to field several weapons that would bring down dragons quickly, effectively and nearly without their own casualties.
Factual error: In the scene when they are traveling at night towards London, when they stop at the blockade of burnt out cars and observe the 'London 666' sign, you can partially see that the sign is meant to be a remaining sign for the A5 road in the UK. They are apparently meant to be in Pembury, which is below south London, whereas the A5 goes north of London. Also Pembury is around 30 miles away from London, not 66.